High-order BDF fully discrete scheme for backward fractional Feynman-Kac equation with nonsmooth data

TL;DR

This paper develops a high-order fully discrete numerical scheme for the backward fractional Feynman-Kac equation with nonsmooth data, achieving up to sixth-order accuracy in time and validating effectiveness through numerical experiments.

Contribution

The paper introduces a novel high-order scheme combining BDF convolution quadrature and finite element methods, overcoming regularity challenges in fractional Feynman-Kac equations.

Findings

Achieves up to 6th-order convergence in time.

Maintains optimal spatial convergence without regularity assumptions.

Numerical experiments confirm the scheme's effectiveness.

Abstract

The Feynman-Kac equation governs the distribution of the statistical observable -- functional, having wide applications in almost all disciplines. After overcoming challenges from the time-space coupled nonlocal operator and the possible low regularity of functional, this paper develops the high-order fully discrete scheme for the backward fractional Feynman-Kac equation by using backward difference formulas (BDF) convolution quadrature in time, finite element method in space, and some correction terms. With a systematic correction, the high convergence order is achieved up to $6$ in time, without deteriorating the optimal convergence in space and without the regularity requirement on the solution. Finally, the extensive numerical experiments validate the effectiveness of the high-order schemes.